In 1868, Baron Albert von Thimus published his two-volume book Die harmonikale Symbolik des Alterthums,
in which he develops, among other things, the Lambdoma-he considered it
to be the rediscovery of a Pythagorean scheme, but it was more likely a
new construction in the Pythagorean spirit. From ancient times we only
know of the Lambdoma's basic form, from which it got its name: when all
whole numbers expressed as fractions, and their reciprocals, are
arranged in two basal series in the form of the Greek letter Λ, this image emerges:

These basal series simultaneously represent the overtone series and its
reflection, the undertone series. If the space between them is filled
out in terms of the overtone series for each undertone, or vice versa,
the result is the complete diagram, which can be continued in both
directions without limit. Image 14 is from Kayser's Lehrbuch der Harmonik, and goes up to index 16; the model exhibited goes up to index 32.

This diagram exhibits various peculiarities which are highly
interesting both musically and symbolically. One of these peculiarities
is that all fractions that have the same tone-value are connected by a
straight line, and all these lines meet at a point which is actually
outside the diagram: the point 0/0. These lines are called equal-tone
lines; the string in the model represents one of these equal-tone lines
in each position. If the string is positioned, for example, at the
value 2/3, it can be seen also to pass through the values 4/6, 6/9,
8/12, etc., which all produce a fifth. At the same time the bridge on
the monochord marks off precisely these proportions, so that the ratio
2/3 is audible as a fifth (the upper part of the string represents the
interval in each case). The second string represents the base tone,
always remaining the same. In this manner all numeric proportions can
be made audible in the model.

Peter Neubäcker

Image 15

Spatial Logarithmic Illustration of the Lambdoma

by Rudolf Stössel

Displayed at the exhibition “Phenomena” in Zurich, 1984

Photo by Dieter Trüstedt

The
structures of the Lambdoma can be further clarified by illustrating the
values that exist in each field in the third dimension. For example,
the values 1/1, 2/1, 3/1, and 4/1 represent the single, doubled,
tripled, and quadrupled string lengths of the monochord, which can then
be built in a model as column heights reaching upwards.

But the model thus emerging does not correspond to the auditory
experience, since for the ear the differences in tones become
continually smaller toward the higher numbers; in other words, the
hearing of the intervals takes place logarithmically. The model shown
here takes these conditions into account.

The
outline of this model is quadratic, like the Lambdoma itself, and the
diagonals of the Lambdoma run directly through the middle of the image
at a constant height. Upon these diagonals lie the values 1/1, 2/2,
3/3, etc., i.e. all values that are identical to the base tone. This
line is called the “generator-tone line.” Starting from it, all
tone-values go upwards and downwards as logarithmic curves.

The Swiss harmonist Rudolf Stössel studied the three-dimensional
illustration of the Lambdoma from various points of view and hearing,
built several models, and published the results in his paper Harmonikale Modelle zur Veranschaulichung der Ober- und Untertonreihe.
At the 1984 exhibition “Phenomena” in Zurich, an especially large
version of this model was displayed, built specially for the
exhibition; this model is shown here.

Image 16

The Tones of the Lambdoma Represented as Tubular Bells

by Dieter and Ulrike Trüstedt

Displayed at the exhibition “Phenomena” in Zurich, 1984

Photo by Dieter Trüstedt

The
musician and physicist Dieter Trüstedt, from Munich, explored several
possibilities of using the structures of the Lambdoma as a musical
instrument. Among them was a framework in the form of the Lambdoma with
index 12 × 12, upon which the intervals that corresponded to the
proportions of the Lambdoma could be electronically generated through
striking.

Another instrument was based on the ch'in, a Chinese stringed
instrument. 13 strings are tuned in the progression of the undertone
series; the overtones for each of these undertones are generated by
striking the strings so that the flageolet tones sound, which are then
electronically amplified. Thus the tones of the Lambdoma can be made
audible up to index 13.

Dieter Trüstedt built a third instrument for the “Phenomena”
exhibition-this is shown in Image 16. Here, the tones of the Lambdoma
are represented by 12 × 12 steel tubular bells; the longest bell is
over 2 meters long. The lengths of the tubular bells do not correspond
to the numbers of the Lambdoma, since the vibrations of bars and bells
obey different laws from those of strings.

Another
form of the Lambdoma as a musical instrument was realized by Peter
Neubäcker in a computer program: here the Lambdoma is shown on the
screen up to a given index-the Lambdoma tone to which the player points
on the screen is then sent to a synthesizer. In this way all tones of
the Lambdoma are available, throughout the entire auditory domain.

The
most charming thing about playing with the Lambdoma is the fact that
musically related tones are always nearby, in contrast with a piano
keyboard, on which the available tones are simply laid out in a row.
Also, in the Lambdoma all the tones are available in their pure tuning,
and thus also the intervals that are formed from the higher prime
numbers, such as 7, 11, 13, etc., which cannot be achieved at all in
our traditional music.

Image 17

Development of the Surfaces of Crystals Represented in the Lambdoma

Above: up to 7 complication steps

Below: up to 12 complication steps

Conceived and drawn by Peter Neubäcker

Image 18

Surface Development of Crystals Represented in the Lambdoma

The Four Quadrants in the Cubic System

Conceived and drawn by Peter Neubäcker

In
the construction of the surfaces of crystals, laws can be established
which can also be expressed in the form of musical intervals in the
realm of harmonics. For example, if a coordinate axis is drawn through
the simplest crystal form, the cube, one surface is parallel to an
axis, the other perpendicular to it. This can be expressed in the
ratios 0/1 = 0 for the perpendicular, 1/0
= ∞ for the parallel surfaces. These two surfaces are described as
primary surfaces. The further appearing surfaces are arranged so that
they form an angle between the two primary surfaces-thus only certain
angles are possible, namely those which always mark off whole numbers
from both axes. So the next possible surface would be that which forms
an angle of 45 degrees to the primary surfaces; since it marks off
equal parts from both axes, it is described as 1/1-other possible
surfaces include 1/2, 2/3, etc. If the axes of the crystal system are
now imagined as two monochord strings placed at a right angles to each
other, then the crystal surfaces always mark off consonant intervals,
and the more consonant the corresponding interval, the more likely the
surface is to occur.

The crystallographer Victor Goldschmidt developed another law which
determines the probability of crystal surfaces-he calls it the Law of
Complication. Thus the following “normal series” appear:

Primary surfaces: A

.

.

.

.

.

.

.

B

N0 = 0

.

.

.

.

.

.

.

∞ = normal series 0.

1st complication: A

.

.

.

C

.

.

.

B

N1 = 0

.

.

.

1

.

.

.

∞ = normal series 1.

2nd complication: A

.

D

.

C

.

E

.

B

N2 = 0

.

1/2

.

1

.

2

.

∞ = normal series 2.

3rd complication: A

F

D

G

C

H

E

I

B

N3 = 0

1/3

1/2

2/3

1

3/2

2

3

∞ = normal series 3.

etc.

The links between the series that follow in each case then emerge
through the adding of the denominator and numerator of the neighboring
fractions. In nature the emerging surfaces only go beyond the 3rd
normal series in rare cases. But the idea of this law of surface
construction can be continued further, leading to the images shown.

The Lambdoma can be understood directly as a coordinate axis in the
cubic crystal system (also for other crystal systems, when the angle of
the axis is changed)-an equal-tone line going from point 0/0 to a
certain tone-value is then identical to the “surface norms,” i.e. the
lines perpendicular to the respective crystal surfaces. According to
Goldschmidt, the “direction of the particle energy that constructs the
surfaces.”

The tone-values lying nearest to the origin of the Lambdoma thus
represent the most commonly appearing crystal surfaces, and the ones
further out represent the less common-but not arbitrarily chosen
tone-values, instead those belonging to the next normal series in each
case, according to Goldschmidt's Law of Complication. In the
illustrations here, all tone-values or surface indices newly appearing
in a normal series are joined by a line-which can be seen most plainly
in the first image. These images can thus be understood as a direct
illustration of Goldschmidt's Law of Complication, or better yet, as
the illustration of the tendency of this law, because such high degrees
of complication only appear rarely in nature.

The similarity in form to certain minerals that grow in a radiant
pattern (for example antimony) is evident, especially when the lower
picture of Image 17 is continued into the spatial realm. It is
theoretically conceivable that a causal connection exists in the sense
of crystallography, in addition to the morphological.

It is interesting to consider which variations from one degree of
complication to the next emerge in the Lambdoma in this illustration. A
process of inversion can be clearly seen here: precisely where the
surface development at the lower degrees reaches farthest into the
Lambdoma, gaps emerge at the higher degrees; so the greatest gap in the
image is at 1/1-that is the direction in which the first surface
emerges. For all further degrees the same phenomenon can be observed.

Following each of the newly emerging numbers and their directions in
the Lambdoma, it becomes apparent that they are the numbers of the
Fibonacci Sequence: the ray reaching the farthest tends to the
direction of the Golden Section, the next is identical in its direction
to that of the “main series of leaf position numbers” described in
Image 8. But because mostly only the lower degrees of complication
appear in nature, here we once again see the polarity of an ideal
tendency towards the Golden Section on the one hand and the realization
through the harmonic intervals on the other hand.